2011年2月11日 · In debate(?), "orthogonal" to mean "not relevant" or "unrelated" also comes from the above meaning. If issues X and Y are "orthogonal", then X has no bearing on Y. If you think of X and Y as vectors, then X has no component in the direction of Y: in other words, it is orthogonal in the mathematical sense.

2015年7月12日 · I have often come across the concept of orthogonality and orthogonal functions e.g in fourier series the basis functions are cos and sine, and they are orthogonal. For vectors being orthogonal means that they are actually perpendicular such that their dot product is zero. However, I am not sure how sine and cosine are actually orthogonal.

2017年8月26日 · An orthogonal basis can be used to decompose something into independent components. For example, the Fourier transform decomposes a time domain function into weights of sines and cosines. A triple in 3D space is a decomposition of a vector in 3D space along 3 orthogonal basis vectors. $\endgroup$ –

2015年8月4日 · Two vectors are orthogonal if their inner product is zero. In other words $\langle u,v\rangle =0$. They are orthonormal if they are orthogonal, and additionally each vector has norm $1$. In other words $\langle u,v \rangle =0$ and $\langle u,u\rangle = \langle v,v\rangle =1$. Example. For vectors in $\mathbb{R}^3$ let

2023年7月30日 · In general, for any matrix, the eigenvectors are NOT always orthogonal. But for a special type of matrix, symmetric matrix, the eigenvalues are always real and eigenvectors corresponding to distinct eigenvalues are always orthogonal. If the eigenvalues are not distinct, an orthogonal basis for this eigenspace can be chosen using Gram-Schmidt.

Generally, those matrices that are both orthogonal and have determinant $1$ are referred to as special orthogonal matrices or rotation matrices. If I read "orthonormal matrix" somewhere, I would assume it meant the same thing as orthogonal matrix. Some examples: $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ is not orthogonal.

So, basically, orthogonal matrix is just a combination of one-dimensional reflectors and rotations written in appropriately chosen orthonormal basis (the coordinate system you're used to, but possibly rotated).

Also, orthogonal set and linearly independent set both generate the same subspace. (Is that right?) Then orthogonal $\rightarrow$ linearly independent but orthogonal $\nleftarrow$ linearly independent is that right? One more question. For T/F, Every orthogonal set is linearly independent (F) Every orthonormal set is linearly independent (T) Why?

The original question was asking about a matrix H and a matrix A, so presumably we are talking about the operator norm.

As far as I know orthogonality is a linear algebraic concept, where for a 2D or 3D case if the vectors are perpendicular we say they are orthogonal. Even it is OK for higher dimensions. But when it comes to random variables I cannot figure out orthogonality.